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anemone
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Let $a,\,b,\,c$ be non-zero real numbers such that $(ab+bc+ca)^3=abc(a+b+c)^3$. Prove that $a,\,b,\,c$ are terms of a geometric sequence.
A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio. The general form of a geometric sequence is a, ar, ar^2, ar^3, ... where a is the first term and r is the common ratio.
To prove a geometric sequence with $(a,b,c)$, you need to show that the ratio between any two consecutive terms is constant. This can be done by dividing the second term by the first term, and then dividing the third term by the second term. If the resulting values are equal, then the sequence is a geometric sequence with a common ratio of b/a. You can continue this process for all terms in the sequence to further support your proof.
Geometric sequences can be found in various natural phenomena, such as the growth of bacteria, population growth, and the spread of diseases. They are also commonly used in financial calculations, such as compound interest and depreciation of assets.
Yes, a geometric sequence can have negative terms. As long as the common ratio is a negative number, the sequence will alternate between positive and negative terms.
A geometric sequence differs from an arithmetic sequence in that the terms are found by multiplying the previous term by a constant value, whereas in an arithmetic sequence, the terms are found by adding a constant value to the previous term. Additionally, in a geometric sequence, the difference between any two consecutive terms is not constant, unlike in an arithmetic sequence.